Classifying Triangles
Theory
A triangle gets two names: one from its sides (scalene, isosceles, equilateral) and one from its angles (acute-angled, right-angled, obtuse-angled). Often you must work out the sides or the angles first. This Year 8 maths page shows how.
By sides. Scalene: no sides equal. Isosceles: two sides equal, and the base angles opposite them are equal; it has one axis of symmetry. Equilateral: three sides equal, so each angle is \(180 \div 3 = 60^\circ\); it has three axes of symmetry and rotational symmetry of order \(3\).
By angles, decided by the largest angle: acute-angled (all angles less than \(90^\circ\)), right-angled (one \(90^\circ\) angle), obtuse-angled (one angle more than \(90^\circ\)).
Naming. \(\triangle ABC\) names the triangle; \(\angle ABC\) names the angle at \(B\).
Same unit first. Convert before you compare: \(60\) mm and \(6\) cm are the same length.
Angles first. Find all three interior angles (angle sum, straight line, exterior angle), then look at the largest. Equal angles give equal sides opposite them, so a triangle can be named isosceles from its angles alone.
One angle of an isosceles triangle: two cases. The known angle is either the angle between the equal sides or one of the base angles. Test both.
On the number plane, count the grid steps along each side, then square and add (Pythagoras’ theorem). Compare the squares.
Symmetry. An axis of symmetry is a fold line whose halves match. The order of rotational symmetry counts how many times a shape lands on itself in one full turn; it is never less than \(1\).
Two names for every triangle:
| Sides | Angles | Symmetry |
|---|---|---|
| scalene: no equal sides | acute: all less than \(90^\circ\) | no axes, order \(1\) |
| isosceles: two equal sides | right: one \(90^\circ\) angle | one axis, order \(1\) |
| equilateral: three equal sides | obtuse: one more than \(90^\circ\) | three axes, order \(3\) |
How to classify a triangle
- Sides: change to one unit (or square the grid steps), then count the equal sides.
- Angles: find all three interior angles, then look at the largest.
- Name it both ways, such as “isosceles and right-angled”.
| \(72 \div 10\) | \(=\) | \(7.2\) |
The sides are \(7.2\) cm, \(7.2\) cm and \(9\) cm. Exactly two are equal, so the triangle is isosceles.
Case (i): \(34^\circ\) is between the equal sides, so \(b + b + 34 = 180\) (angle sum of a triangle).
Case (ii): \(34^\circ\) is a base angle, so the other base angle is also \(34^\circ\).
| \(2b\) | \(=\) | \(146\) |
| \(b\) | \(=\) | \(73\) |
| \(180 - 34 - 34\) | \(=\) | \(112\) |
Case (i): \(73^\circ\) and \(73^\circ\), acute-angled.
Case (ii): \(34^\circ\) and \(112^\circ\), obtuse-angled.
(i) Classify it by its sides.
(ii) Is it right-angled?
| \(PQ^2\) | \(=\) | \(5^2 = 25\) |
| \(QR^2\) | \(=\) | \(3^2 + 4^2 = 25\) |
| \(PR^2\) | \(=\) | \(2^2 + 4^2 = 20\) |
(i) \(PQ = QR\) and \(PR\) differs, so it is isosceles.
(ii) \(25 + 20 = 45\), not \(25\): no two squares add to the third, so it is not right-angled.
(i) \(m\)
(ii) the perimeter
| \(3m - 4\) | \(=\) | \(m + 8\) |
| \(2m\) | \(=\) | \(12\) |
| \(m\) | \(=\) | \(6\) |
| \(14 + 14 + 7\) | \(=\) | \(35\) |
(i) \(m = 6\): the equal sides are \(14\) cm and the third is \(7\) cm.
(ii) The perimeter is \(35\) cm.
\(\angle ACB = 180 - 105 = 75^\circ\) (angles on a straight line), so \(\angle B = 180 - 75 - 75 = 30^\circ\) (angle sum of a triangle).
Two angles are \(75^\circ\), so the sides opposite them are equal (\(AB = BC\)): the triangle is isosceles. Its largest angle is \(75^\circ\), so it is acute-angled.
Common pitfalls
Frequently asked questions
What is a scalene triangle?
A triangle with no equal sides and no equal angles.
What is an isosceles triangle?
A triangle with two equal sides; the angles opposite those sides are equal too.
What is an equilateral triangle?
A triangle with three equal sides and three angles of \(60^\circ\).
How do you classify a triangle by its angles?
Look at the largest angle: less than \(90^\circ\) is acute-angled, exactly \(90^\circ\) is right-angled, more than \(90^\circ\) is obtuse-angled.
How many axes of symmetry does an equilateral triangle have?
Three, and its order of rotational symmetry is \(3\).